The chi-square test involves using the chi-square
distribution
to approximate the underlying exact distribution.
The approximation becomes better as the
expected cell frequencies
grow larger, and may be inappropriate
for tables with very small expected cell frequencies.
For tables with expected cell frequencies less than 5,
the chi-square approximation may not be reliable.
A standard (and conservative)
rule of thumb is to avoid using
the chi-square test for tables with expected
cell frequencies less than 1, or when more than 20% of
the table cells have expected cell frequencies
less than 5.
Another rule of thumb (due to
Roscoe and Byars) is that
the average expected cell frequency should
be at least 1 when the expected cell frequencies
are close to equal, and 2 when they are not.
(If the chosen
significance level
is 0.01 instead of 0.05, then double these numbers.)
Koehler and Larntz suggest that if the total number
of observations is at least 10, the number categories
is at least 3, and the square of the total number
of observations is at least 10 times the number of
categories, then the chi-square approximation
should be reasonable.
The table of expected values will reveal whether any
of these conditions is true, and Prophet will also generate
an appropriate warning in the test results.
The table of contributions to chi-square gives the value for
each cell
of the square of the difference between the observed and expected
values for the cell, divided by the
expected value for the cell.
The standardized residuals are the signed square roots
of these values. Positive residuals indicate that
the observed cell frequency is greater than the expected
cell frequency, and
negative residuals indicate that
the observed cell frequency is less than the expected
cell frequency.
If there are standardized residuals greater than 2 or
less than -2, those cells are not being fitted very
well by the hypothesized distribution. A large residual
may also mean that a particular cell is an
outlier.
When the categories have a natural order, then a pattern
to residuals (e.g., large negative ones at one end of
the table, with large positive ones at the other end
of the table) may indicate the possibility of an
implicit factor.
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